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A110344 a(n) = Sum_{k=0..n-1} (n+k) = n(3n-1)/2 if n is even; a(n) = Sum_{k=0..n-1} (n-k) = n(n+1)/2 if n is odd.

%I A110344 #23 Sep 11 2022 06:21:36
%S A110344 1,5,6,22,15,51,28,92,45,145,66,210,91,287,120,376,153,477,190,590,
%T A110344 231,715,276,852,325,1001,378,1162,435,1335,496,1520,561,1717,630,
%U A110344 1926,703,2147,780,2380,861,2625,946,2882,1035,3151,1128,3432,1225,3725,1326
%N A110344 a(n) = Sum_{k=0..n-1} (n+k) = n(3n-1)/2 if n is even; a(n) = Sum_{k=0..n-1} (n-k) = n(n+1)/2 if n is odd.
%H A110344 Colin Barker, <a href="/A110344/b110344.txt">Table of n, a(n) for n = 1..1000</a>
%H A110344 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F A110344 From _Emeric Deutsch_, Aug 01 2005: (Start)
%F A110344 a(2n+1) = A000217(2n+1) = (n+1)(2n+1) (triangular numbers with odd index).
%F A110344 a(2n) = A000326(2n) = A049452(n) = n(6n-1) (pentagonal numbers with even index).
%F A110344 (End)
%F A110344 a(n) = n*( 2*n + (n-1)*(-1)^n )/2. - _Luce ETIENNE_, Jul 08 2014
%F A110344 From _Colin Barker_, Feb 17 2015: (Start)
%F A110344 a(n) = 3*a(n-2)-3*a(n-4)+a(n-6).
%F A110344 G.f.: -x*(7*x^3+3*x^2+5*x+1) / ((x-1)^3*(x+1)^3). (End)
%F A110344 Sum_{n>=1} 1/a(n) = 4*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2. - _Amiram Eldar_, Sep 11 2022
%e A110344 a(3) = 3 + 2 +1 = 6.
%e A110344 a(6) = 6 + 7 + 8 + 9 + 10 + 11 = 51.
%p A110344 a:=proc(n) if n mod 2=0 then n*(3*n-1)/2 else n*(n+1)/2 fi end: seq(a(n),n=1..60); # _Emeric Deutsch_
%t A110344 a[n_] := n*(2*n + (n - 1)*(-1)^n)/2; Array[a, 50] (* _Amiram Eldar_, Sep 11 2022 *)
%o A110344 (PARI) Vec(-x*(7*x^3+3*x^2+5*x+1)/((x-1)^3*(x+1)^3) + O(x^100)) \\ _Colin Barker_, Feb 17 2015
%Y A110344 Cf. A000217, A000326, A049452.
%K A110344 nonn,easy
%O A110344 1,2
%A A110344 _Amarnath Murthy_, Jul 20 2005
%E A110344 More terms from _Emeric Deutsch_, Aug 01 2005